Cosine: Root, what are you doing?

Root: I thought... wait, what's this session about?

Cosine: Chopping at the Gap. Between prime numbers.

Root: Oh, chopping! I heard something else.

IT WAS A PRIME DIRECTIVE |

Maud: We're already rolling.

Cosine: Gah! Okay. Big news about...

*...*

Maud: We couldn't get a feed to Sine either. Sorry.

Cosine: ...about twin primes. Sigh. I miss my twin.

Secant: I miss mine too.

ArcCos: At least yours haven't gone insane...

Sexi: Well, if no twins are available, I'll join Cosine.

Cosine: Really? Under what authority??

Sexi: The sexy primes. Primes separated by six, like 5 and 11, or 7 and 13.

Cosine: I feel like you're making that term up.

Sexi: Nope. There's also sexy foursomes.

Maud: Okaaay. Uh, back up to twin primes for a second, ladies. There's a song people can watch! It should get everyone up to speed.

Cosine: Indeed. Now then, if everyone is on board with twin - and sexy - primes, let's build up towards the most recent breakthrough.

Sexi: Maud's video referenced Goldston, who worked with Cem Yildirim and Janos Pintz in 2005, on what has become known as GPY. Interestingly, Pintz actually came on board last, after an error was found in the initial GY proof.

Cosine: But we can see how a math proof called GYP isn't the best idea...

Sexi: GPY's efforts regarding the smallest possible prime gap didn't close the door to infinity. Thus we fast forward to April 17, 2013, when Yitang Zhang submits a paper to Annals of Mathematics.

Cosine: The editors saw there was something to it, and fast tracked it through. REALLY fast tracked it. By May 13th, Zhang was scheduled to give a seminar at Harvard.

Sexi: In the following link, you can see how the news was getting out even on May 12th. Visit the Not Even Wrong blog, in the post "Number Theory News".

Cosine: Check out the comments there too; the speaker was listed as "Tom Zhang". Now, for a summary of the actual presentation, check out the n-Category Cafe here.

Sexi: If links don't thrill you though, in brief, Zhang showed that there are infinitely many primes that differ by N. Where N is a number smaller than 70 million.

Secant: Let's add some context there.

Sexi: POINT ONE. We already knew there was NO upper bound. In other words N can be a number as large as you want. Mr. Honner provides a very simple proof of this fact, in his Infinite Prime Gaps post from May 15.

Cosine: In other words, you want a prime, then over 1,000,000,000,000 composite numbers, then the next prime... sure, we can do that.

Sexi: POINT TWO. We already know that the gaps get bigger as you count more and more numbers. In fact, the now hundred year old "Prime Number Theorem" states that among the first P numbers, about P/(ln P) are prime.

Cosine: Meaning among the first 100 numbers, 100/(ln 100) or about 22 are prime...

Expona: Actual value 25 primes!

Cosine: Among the first 1000 numbers, 1000/(ln 1000) or about 145 are prime...

Expona: Though the real value is 168 primes!

Cosine: Look, the formula is merely asymptotically approaching the function pi(x), so hush up.

Expona: You said log base e, I'm just filling in for Logan!

Sexi: POINT THREE. Given the gaps get bigger, and there is no upper bound on them - how can it be that you suddenly have two primes nearly side by side? Mind blown!

Cosine: Assuming you define "nearly" as 70 million.

Secant: That said, according to the latest Numberphile video, Gaps between Primes, people are already working on knocking the gap number even lower.

Maud: Now, there's a human element here too.

Cosine: Do tell?

Maud: Many news reports point out how Zhang was a relative unknown in the mathematics community. Moreover, he made the breakthrough by applying existing methods, those of GPY. So in a way, there's hope for everyone.

Sexi: Agreed. The idea even came to Zhang while taking a break from years of work. So sometimes the best thing you can do is take a break!

Maud: By the way, my information is from Simons Science News from May 19th: Unheralded Mathematician Bridges the Prime Gap. Many nice quotes there, like "There are a lot of chances in your career, but the important thing is to keep thinking."

Cosine: There's also an interesting writeup connecting primes to randomness in Slate from May 22nd! The Beauty of Bounded Gaps.

Sexi: And, to round things out, the prime gap news was in the New York Times. So now everyone knows about it. With that, I think we're done here!

*...*

Root: So, does anyone feel like going to a mall?